The Saudi-led coalition reported casualties early on 6 June after repulsing a second ground attack by Houthi rebels at the border, coming just a day after both the warring sections had agreed to attend a UN-brokered peace plan.
"The Saudi armed forces today were able to repel an attack from the Yemeni side targeting several locations in Jazan and Najran," a coalition statement carried by the official Saudi Press Agency said, noting that four Saudis and many rebels were killed.
At least 21 people have died in the ongoing ground combat since 4 June in the Aden area, while air raids continued against Daleh and Shabwa provinces.
Houthi rebels and Yemen's exiled government have agreed to attend the peace talks in Geneva, aimed at ending the 10-week conflict that has cost more than 2,000 lives.
The UN meeting, set for 14 June, will aim at a ceasefire, a Houthi withdrawal plan and deliveries of humanitarian aid.
UN peace talks
It will be the first effort to stop the fighting which began ten weeks ago with a Saudi-led coalition bombing rebel targets in impoverished Yemen.
"We accepted the invitation of the United Nations to go to the negotiating table in Geneva without preconditions," said Daifallah al-Shami, a senior member of the rebels' political wing.
Ezzedine al-Isbahi, information minister of the Yemeni government exiled in the Saudi capital, said the government would also send a delegation to the meet, reports AFP.
He said the meeting would involve consultations on implementing Resolution 2216, which the Security Council passed in April, imposing an arms embargo on the Houthi rebels and demanding them to relinquish seized territory.
The war has forced more than half a million people from their homes and placed 20 million in need of aid, says the UN which described Yemen's humanitarian crisis as "catastrophic".
The Saudi-led coalition began its air campaign on 25 March after the Houthi rebels, who captured capital Sanaa in September closed in on the southern port city of Aden and forced the Saudi-backed Hadi to flee to Riyadh.
Saudi Arabia sees the rebels as proxies for regional Shiite powerhouse Iran seeking to destabilise the region.